Game representations of classes of piecewise definable functions

TL;DR

This paper introduces a unified framework of reduction games on 8 that represent various classes of piecewise definable functions, aiding the study of reducibilities in descriptive set theory.

Contribution

It develops a general method to construct reduction games for different classes of functions, including piecewise, limit, and 3-measurable functions, enhancing combinatorial analysis tools.

Findings

Constructed games for piecewise functions

Developed games for pointwise limits of functions

Created games for 3-measurable functions

Abstract

We present a general way of defining various reduction games on \omega\ which "represent" corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for \Gamma-measurable functions. These games turn out to be useful as a combinatorial tool for the study of general reducibilities for subsets of the Baire space [10].